Essential Dimension and the Flats Spanned by a Point Set
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Let P be a finite set of points in ℝd or ℂd.We answer a question of Purdy on the conditions under which the number of hyperplanes spanned by P is at least the number of (d−2)-flats spanned by P.
In answering this question, we define a new measure of the degeneracy of a point set with respect to affine subspaces, termed the essential dimension. We use the essential dimension to give an asymptotic expression for the number of k-flats spanned by P, for 1≤k≤d−1.
Mathematics Subject Classification (2000)52C10
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